# nLab (infinity,1)-category of (infinity,1)-functors

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## Idea

The generalization of the notion of functor category from category theory to (∞,1)-higher category theory.

## Definition

Let $C$ and $D$ be (∞,1)-categories, taken in their incarnation as quasi-categories. Then

$\mathrm{Func}\left(C,D\right):=\mathrm{sSet}\left(C,D\right)$Func(C,D) := sSet(C,D)

is the simplicial set of morphisms of simplicial sets between $C$ and $D$ (in the standard sSet-enrichment of $\mathrm{SSet}$):

$\mathrm{sSet}\left(C,D\right):=\left[C,D\right]:=\left(\left[n\right]↦{\mathrm{Hom}}_{\mathrm{sSet}}\left(\Delta \left[n\right]×C,D\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D)) \,.

The objects in $\mathrm{Fun}\left(C,D\right)$ are the (∞,1)-functors from $C$ to $D$, the morphisms are the corresponding natural transformations or homotopies, etc.

###### Proposition

The simplicial set $\mathrm{Fun}\left(C,D\right)$ is indeed a quasi-category.

In fact, for $C$ and $D$ any simplicial sets, $\mathrm{Fun}\left(C,D\right)$ is a quasi-category if $D$ is a quasi-category.

###### Proof

Using that sSet is a closed monoidal category the horn filling conditions

$\begin{array}{ccc}\Lambda \left[n{\right]}_{i}& \to & \left[C,D\right]\\ ↓& ↗\\ \Delta \left[n\right]\end{array}$\array{ \Lambda[n]_i &\to& [C,D] \\ \downarrow & \nearrow \\ \Delta[n] }

are equivalent to

$\begin{array}{ccc}C×\Lambda \left[n{\right]}_{i}& \to & D\\ ↓& ↗\\ C×\Delta \left[n\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C \times \Lambda[n]_i &\to& D \\ \downarrow & \nearrow \\ C \times \Delta[n] } \,.

Here the vertical map is inner anodyne for inner horn inclusions $\Lambda \left[n{\right]}_{i}↪\Delta \left[n\right]$, and hence the lift exists whenever $D$ has all inner horn fillers, hence when $D$ is a quasi-category.

For the definition of $\left(\infty ,1\right)$-functors in other models for $\left(\infty ,1\right)$-categories see (∞,1)-functor.

## Properties

### Models

The projective and injective global model structure on functors as well as the Reedy model structure if $C$ is a Reedy category presents $\left(\infty ,1\right)$-categories of $\left(\infty ,1\right)$-functors, at least when there exists a combinatorial simplicial model category model for the codomain.

Let

Write $N:\mathrm{sSet}\mathrm{Cat}\to \mathrm{sSet}$ for the homotopy coherent nerve. Since this is a right adjoint it preserves products and hence we have a canonical morphism

$N\left(C\right)×N\left(\left[C,A\right]\right)\simeq N\left(C×\left[C,A\right]\right)\stackrel{N\left(\mathrm{ev}\right)}{\to }N\left(A\right)$N(C) \times N([C,A]) \simeq N(C \times [C,A]) \stackrel{N(ev)}{\to} N(A)

induced from the hom-adjunct of $\mathrm{Id}:\left[C,A\right]\to \left[C,A\right]$.

The fibrant-cofibrant objects of $\left[C,A\right]$ are enriched functors that in particular take values in fibrant cofibrant objects of $A$. Therefore this restricts to a morphism

$N\left(C\right)×N\left(\left[C,A{\right]}^{\circ }\right)\stackrel{{N}_{\mathrm{hc}}\left(\mathrm{ev}\right)}{\to }N\left({A}^{\circ }\right)\phantom{\rule{thinmathspace}{0ex}}.$N(C) \times N([C,A]^\circ) \stackrel{N_{hc}(ev)}{\to} N(A^\circ) \,.

By the internal hom adjunction this corresponds to a morphism

$N\left(\left[C,A{\right]}^{\circ }\right)\stackrel{}{\to }\mathrm{sSet}\left({N}_{\mathrm{hc}}\left(C\right),N\left({A}^{\circ }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$N([C,A]^\circ) \stackrel{}{\to} sSet(N_{hc}(C), N(A^\circ)) \,.

Here ${A}^{\circ }$ is Kan complex enriched by the axioms of an ${\mathrm{sSet}}_{\mathrm{Quillen}}$- enriched model category, and so $N\left({A}^{\circ }\right)$ is a quasi-category, so that we may write this as

$\cdots =\mathrm{Func}\left(N\left(C\right),N\left({A}^{\circ }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots = Func(N(C), N(A^\circ)) \,.
###### Proposition

This canonical morphism

$N\left(\left[C,A{\right]}^{\circ }\right)\stackrel{}{\to }\mathrm{Func}\left(N\left(C\right),N\left({A}^{\circ }\right)\right)$N([C,A]^\circ) \stackrel{}{\to} Func(N(C), N(A^\circ))

is an $\left(\infty ,1\right)$-equivalence in that it is a weak equivalence in the model structure for quasi-categories.

This is (Lurie, prop. 4.2.4.4).

###### Proof

The strategy is to show that the objects on both sides are exponential objects in the homotopy category of ${\mathrm{sSet}}_{\mathrm{Joyal}}$, hence isomorphic there.

That $\mathrm{Func}\left(N\left(C\right),N\left({A}^{\circ }\right)\right)\simeq \left(N\left({A}^{\circ }\right){\right)}^{N\left(C\right)}$ is an exponential object in the homotopy category is pretty immediate.

That the left hand is an isomorphic exponential follows from (Lurie, corollary A.3.4.12), which asserts that for $C$ and $D$ $\mathrm{sSet}$-enriched categories with $C$ cofibrant and $A$ as above, we have that composition with the evaluation map induces a bijection

${\mathrm{Hom}}_{\mathrm{Ho}\left(\mathrm{sSet}\mathrm{Cat}\right)}\left(D,\left[C,A{\right]}^{\circ }\right)\stackrel{\simeq }{\to }{\mathrm{Hom}}_{\mathrm{Ho}\left(\mathrm{sSet}\mathrm{Cat}\right)}\left(C×D,{A}^{\circ }\right)\phantom{\rule{thinmathspace}{0ex}}.$Hom_{Ho(sSet Cat)}(D, [C,A]^\circ) \stackrel{\simeq}{\to} Hom_{Ho(sSet Cat)}(C \times D, A^\circ) \,.

Since $\mathrm{Ho}\left(\mathrm{sSet}{\mathrm{Cat}}_{\mathrm{Bergner}}\right)\simeq \mathrm{Ho}\left({\mathrm{sSet}}_{\mathrm{Joyal}}\right)$ this identifies also $N\left(\left[C,A{\right]}^{\circ }\right)$ with the exponential object in question.

### Limits and colimits

For $C$ an ordinary category that admits small limits and colimits, and for $K$ a small category, the functor category $\mathrm{Func}\left(D,C\right)$ has all small limits and colimits, and these are computed objectwise. See limits and colimits by example. The analogous statement is true for $\left(\infty ,1\right)$-categories of $\left(\infty ,1\right)$-functors

###### Propositon

Let $K$ and $C$ be quasi-categories, such that $C$ has all colimits indexed by $K$.

Let $D$ be a small quasi-category. Then

• The $\left(\infty ,1\right)$-category $\mathrm{Func}\left(D,C\right)$ has all $K$-indexed colimits;

• A morphism ${K}^{▹}\to \mathrm{Func}\left(D,C\right)$ is a colimiting cocone precisely if for each object $d\in D$ the induced morphism ${K}^{▹}\to C$ is a colimiting cocone.

This is (Lurie, corollary 5.1.2.3).

### Equivalences

###### Proposition

A morphism $\alpha$ in $\mathrm{Func}\left(D,C\right)$ (that is, a natural transformation) is an equivalence if and only if each component ${\alpha }_{d}$ is an equivalence in $C$.

This is due to (Joyal, Chapter 5, Theorem C).

## Reference

The intrinsic definition is in section 1.2.7 of

The discussion of model category models is in A.3.4.

The theorem about equivalences is in

Revised on May 16, 2013 01:04:10 by Urs Schreiber (89.204.137.135)